Difference between revisions of "Manuals/calci/HESSENBERG"

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*To be exact, an upper Hessenberg matrix has zero entries below the first subdiagonal, and a lower Hessenberg matrix has zero entries above the first superdiagonal.  
 
*To be exact, an upper Hessenberg matrix has zero entries below the first subdiagonal, and a lower Hessenberg matrix has zero entries above the first superdiagonal.  
 
*Here MATRIX("hessenberg") displays the hessenberg matrix of order 3.  
 
*Here MATRIX("hessenberg") displays the hessenberg matrix of order 3.  
*An nxn matrix with <math>a_{i,j}=0</math> for <math>i\ge j+1</math> is called a Hessenberg matrix.
+
*An nxn matrix with <math>a_{i,j}=0</math> for <math>i\gg j+1</math> is called a Hessenberg matrix.
 
*So the form of a Hessenberg matrix is: \begin{bmatrix}
 
*So the form of a Hessenberg matrix is: \begin{bmatrix}
 
a_{11} & a_{12} & a_{13} \cdots & a_{1(n-1)}& a_{1n}    \\
 
a_{11} & a_{12} & a_{13} \cdots & a_{1(n-1)}& a_{1n}    \\

Revision as of 13:13, 24 April 2015

MATRIX("HESSENBERG",order)


  • is the order of the Hessenberg matrix.

Description

  • This function gives the matrix with the property of Hessenberg matrix.
  • A Hessenberg matrix is a special kind of square matrix, one that is "almost" triangular.
  • To be exact, an upper Hessenberg matrix has zero entries below the first subdiagonal, and a lower Hessenberg matrix has zero entries above the first superdiagonal.
  • Here MATRIX("hessenberg") displays the hessenberg matrix of order 3.
  • An nxn matrix with for is called a Hessenberg matrix.
  • So the form of a Hessenberg matrix is: \begin{bmatrix}

a_{11} & a_{12} & a_{13} \cdots & a_{1(n-1)}& a_{1n} \\ a_{21} & a_{22} & a_{23} \cdots& a_{2(n-1)}& a_{2n} \\ 0 & a_{32} & a_{33} \cdots& a_{3(n-1)}& a_{3n} \\ 0 & 0 & a_{43} \cdots& a_{4(n-1)}& a_{4n} \\ 0 & 0 & 0 \cdots& a_{5(n-1)}& a_{5n} \\ \vdots & \ddots & \vdots \\ 0 & 0 & 0 & a_{(n-1)(n-2)} & a_{(n-1)(n-1)} & a_{(n-1)n}\\ 0 & 0 & 0 & 0 & a_{n(n-1)} & a_{nn}\\ \end{bmatrix}

  • So the matrix is zero below the first subdiagonal.
  • If the matrix is symmetric or Hermitian, the form is tridiagonal.

Examples

  • 1.MATRIX("hessenberg")
-10 -50 -92
-32 62 33
0 -99 -81
  • 2.MATRIX("hessenberg",6)
99 88 -73 20 -17 -10
-28 40 -2 15 -48 55
0 -46 56 -76 -85 -70
0 0 12 -72 72 -17
0 0 0 -67 -26 -6
0 0 0 0 -13 50

See Also

References